The bridge on Route 70 near the state fairgrounds in Little Rock, Ark., has no shoulder for its four narrow lanes and has been rated "structurally deficient." Dating to the 1930s, the bridge was due to be replaced this year, but the project was put on hold as Arkansas, like many other states, grapples with an imminent problem: the federal Highway Trust Fund could soon run out of money.

The dwindling fund, the main source of federal cash to build and maintain roads and transit systems, could mean a delay in...